login
O.g.f.: exp( Sum_{n>=1} (sigma(2*n)^2 - sigma(n)^2) * x^n/n ).
1

%I #6 Jul 24 2013 23:30:27

%S 1,8,52,288,1396,6208,25744,100608,374500,1336488,4596000,15297056,

%T 49444368,155640640,478268800,1437600000,4234216836,12238666208,

%U 34761065924,97130259232,267280386128,724987680384,1940011007056,5125212451584,13376644454672,34512562565224

%N O.g.f.: exp( Sum_{n>=1} (sigma(2*n)^2 - sigma(n)^2) * x^n/n ).

%C Compare to the Jacobi theta_3 function:

%C 1 + 2*Sum_{n>=1} x^(n^2) = exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).

%C Here sigma(n) = A000203(n), the sum of the divisors of n.

%F Logarithmic derivative yields A227733.

%e G.f.: A(x) = 1 + 8*x + 52*x^2 + 288*x^3 + 1396*x^4 + 6208*x^5 + 25744*x^6 +...

%e where

%e log(A(x)) = 8*x + 40*x^2/2 + 128*x^3/3 + 176*x^4/4 + 288*x^5/5 + 640*x^6/6 +...+ A227733(n)*x^n/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n,(sigma(2*m)^2-sigma(m)^2)*x^m/m)+x^2*O(x^n)), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A227733.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 24 2013